1. The problem statement, all variables and given/known data A rod of leangth L and flexural rigidity EI is pinned at one end by means of torsional spring having a constant beta, and fixed on the other end. The rod is subjected to a compressive axial force P. Determine the critical load for instability.(image of the problem is attached) 2. Relevant equations As far as I understand, the way to solve this problem is: M(x)=-R*x-P*v(x)+beta*v'(0) , where R is the reaction on the left end at the y direction v(x) is the deflection function and v'(x) is the angle function. M(x)=v''(x)*IE 3. The attempt at a solution the two equations above yields: v''(x)+(P/EI)*v(x)= (beta*v'(0)-R*x)/EI -> v(x)=A*sin(sqrt(P/EI)*x)+B*cos(sqrt(P/EI)*x)+beta*v'(0)-R*x to find those constants the boundry conditions are: v(o)=0 v(L)=0 v'(L)=0 How do I represent v'(0) which is unknown ?